Relation Between Zeros and Coefficients MCQs Quiz | Class 10
This quiz is designed for Class X students, covering the subject Mathematics (Code 041) under Unit II: Algebra. It focuses on the topic of “Relation Between Zeros and Coefficients,” including concepts like quadratic polynomials, sum and product of zeros, and their practical applications. Test your understanding by attempting these Multiple Choice Questions and download a detailed answer PDF for review.
Relation Between Zeros and Coefficients: A Detailed Guide
Understanding the relationship between the zeros (roots) of a polynomial and its coefficients is fundamental in algebra. For quadratic polynomials, this relationship provides a powerful tool to analyze and construct polynomial equations without explicitly finding the zeros. This guide will delve into these concepts relevant for Class 10 Mathematics.
What is a Quadratic Polynomial?
A polynomial of degree 2 is called a quadratic polynomial. Its general form is P(x) = ax^2 + bx + c, where ‘a’, ‘b’, and ‘c’ are real numbers and ‘a’ is not equal to 0. The zeros of a quadratic polynomial are the values of ‘x’ for which P(x) = 0. A quadratic polynomial has at most two zeros.
Relationship Between Zeros and Coefficients
Let alpha (α) and beta (β) be the two zeros of a quadratic polynomial P(x) = ax^2 + bx + c. There are two key relationships:
- Sum of Zeros: The sum of the zeros is equal to the negative of the coefficient of x divided by the coefficient of x^2.
α + β = -b/a - Product of Zeros: The product of the zeros is equal to the constant term divided by the coefficient of x^2.
α * β = c/a
Forming a Quadratic Polynomial When Zeros are Given
If the zeros, α and β, of a quadratic polynomial are known, the polynomial can be constructed using the formula:
P(x) = k [x^2 – (α + β)x + αβ]
Where ‘k’ is any non-zero real number. Often, for simplicity, k is taken as 1, leading to:
P(x) = x^2 – (Sum of Zeros)x + (Product of Zeros)
Applications of Zeros and Coefficients
- Finding Unknown Coefficients: If one or more zeros are given, or if the sum/product of zeros is known, we can use these relations to find unknown coefficients in the polynomial.
- Analyzing the Nature of Zeros: The signs of the sum and product of zeros can give insights into the signs of the individual zeros. For example, if the product is positive, both zeros have the same sign (both positive or both negative). If the product is negative, the zeros have opposite signs.
- Solving Related Problems: Complex expressions involving zeros (like α^2 + β^2 or 1/α + 1/β) can often be simplified and evaluated using the sum and product formulas, without needing to find the individual zeros.
Example Table
| Polynomial | a | b | c | Sum of Zeros (α+β = -b/a) | Product of Zeros (αβ = c/a) |
|---|---|---|---|---|---|
| x^2 – 7x + 10 | 1 | -7 | 10 | -(-7)/1 = 7 | 10/1 = 10 |
| 2x^2 + 5x – 3 | 2 | 5 | -3 | -5/2 | -3/2 |
Quick Revision Checklist
- General form of quadratic polynomial: ax^2 + bx + c (a ≠ 0).
- Sum of zeros (α + β) = -b/a.
- Product of zeros (αβ) = c/a.
- Polynomial formation: x^2 – (sum)x + (product).
- Know how to use these relations to find missing coefficients.
Extra Practice Questions
- If α and β are the zeros of the polynomial 4x^2 – 5x – 1, then find the value of α + β.
Answer: 5/4 - Find a quadratic polynomial whose zeros are -2 and 3.
Answer: x^2 – x – 6 - If one zero of the quadratic polynomial (k-1)x^2 + kx + 1 is -3, find the value of k.
Answer: 4/3 - If α and β are the zeros of x^2 + 6x + 2, find the value of 1/α + 1/β.
Answer: -3 - For the polynomial x^2 – 8x + k, if the product of zeros is 12, what is the value of k?
Answer: 12
Content created and reviewed by the CBSE Quiz Editorial Team based on the latest NCERT textbooks and CBSE syllabus. Our goal is to help students practice concepts clearly, confidently, and exam-ready through well-structured MCQs and revision content.